Critical and noncritical regions on the critical hyperbola

Melo, Jéssyca Lange Ferreira; Santos, Ederson Moreira dos

doi:10.1007/978-3-319-19902-3_21

Cited by 4 publications

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“…The precise condition on the size of λ µ r (for the case with rs = 1) is specified at (2.2.1) and (2.2.2) ahead. Actually, such condition appeared before in (MELO; SANTOS, 2015) and corresponds to the hypothesis λ < λ 1 for equation (1.0.2). Moreover, as proved in (VORST, 1992, Theorem 4.2), if such condition is not verified, then (HS) may have no positive solution in star-shaped domains.…”

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confidence: 86%

On Hamiltonian systems with critical Sobolev exponents

Guimarães

Santos

2023

Journal of Differential Equations

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confidence: 86%

On Hamiltonian systems with critical Sobolev exponents

Guimarães

Santos

2023

Journal of Differential Equations

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“…Over the past 30 years, Hamiltonian systems have been widely studied with results including, but not limited to, existence, multiplicity, concentration phenomena, positivity, symmetry, and Liouville theorems. We redirect the interested reader to the surveys [12,29,70] for an overview of the topic and to the works [13,19,20,41] for some recent results. One of the first mathematical works studying systems of Hardy-Hénon type equations were done by Calanchi and Ruf in [18].…”

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confidence: 99%

A Variational Approach Towards Supercritical Hamiltonian Systems and (p, 2)-Laplace Equations

Wong

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The aim of the dissertation is to use a variational principle on convex subsets of a Banach space to study two types of nonlinear partial differential equations. We are interested in studying elliptic problems with supercritical nonlinearities as we lose the compactness property, resulting to the possibility of standard variational (p, 2)-Laplace Operator 5 Systems of Nonhomogeneous Nonlinear Elliptic Equations on a Bounded Annular Domain 5.1 Existence of a Positive Solution via a Variational Formulation . .

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Existence of Solutions on the Critical Hyperbola for a Pure Lane–Emden System with Neumann Boundary Conditions

Pistoia

Schiera

Tavares

2023

International Mathematics Research Notices

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We study the following Lane–Emden system: $$\begin{align*} & -\Delta u=|v|^{q-1}v \quad \ \textrm{in}\ \Omega, \qquad -\Delta v=|u|^{p-1}u \quad \ \textrm{in}\ \Omega, \qquad u_{\nu}=v_{\nu}=0 \quad \ \textrm{on}\ \partial \Omega, \end{align*}$$with $\Omega $ a bounded regular domain of ${\mathbb{R}}^{N}$, $N \ge 4$, and exponents $p, q$ belonging to the so-called critical hyperbola $1/(p+1)+1/(q+1)=(N-2)/N$. We show that, under suitable conditions on $p, q$, least-energy (sign-changing) solutions exist, and they are classical. In the proof we exploit a dual variational formulation, which allows to deal with the strong indefinite character of the problem. We establish a compactness condition which is based on a new Cherrier-type inequality. We then prove such condition by using as test functions the solutions to the system in the whole space and performing delicate asymptotic estimates. If $N \ge 5$, $p=1$, the system above reduces to a biharmonic equation, for which we also prove existence of least-energy solutions. Finally, we prove some partial symmetry and symmetry-breaking results in the case $\Omega $ is a ball or an annulus.

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Critical and noncritical regions on the critical hyperbola

Cited by 4 publications

References 28 publications

On Hamiltonian systems with critical Sobolev exponents

On Hamiltonian systems with critical Sobolev exponents

A Variational Approach Towards Supercritical Hamiltonian Systems and (p, 2)-Laplace Equations

Existence of Solutions on the Critical Hyperbola for a Pure Lane–Emden System with Neumann Boundary Conditions

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